Fisika Listrik, Gelombang, dan
Cahaya
Ir. Hadha Afrisal, S.T., M.Sc., IPP
Materi
Textbook
Vector vs. Scalar Review
A library is located 0.5 mile from you.
Can you point where exactly it is?
You also
need to
know the
direction in
which you
should
walk to the
library!
• All physical quantities encountered in this text will be either a scalar or a vector
• A vector quantity has both magnitude (value + unit) and direction
• A scalar is completely specified by only a magnitude (value + unit)
Vector and Scalar Quantities
❑ Vectors ❑ Scalars:
◼ Displacement ◼ Distance
◼ Velocity (magnitude and ◼ Speed (magnitude of
direction!) velocity)
◼ Acceleration ◼ Temperature
◼ Force ◼ Mass
◼ Momentum ◼ Energy
◼ Time
To describe a vector we need more information than to describe a
scalar! Therefore vectors are more complex!
Important Notation
❑ To describe vectors we will use:
◼ The bold font: Vector A is A
◼ Or an arrow above the vector: A
◼ In the pictures, we will always show
vectors as arrows
◼ Arrows point the direction
◼ To describe the magnitude of a
vector we will use absolute value
sign: A or just A,
◼ Magnitude is always positive, the
magnitude of a vector is equal to
the length of a vector.
Properties of Vectors
• Equality of Two Vectors
• Two vectors are equal if they have the same
magnitude and the same direction
• Movement of vectors in a diagram
• Any vector can be moved parallel to itself
without being affected
❑ Negative Vectors
◼ Two vectors are negative if they have the same
magnitude but are 180° apart (opposite directions)
( )
A = −B; A + − A = 0 A
B
Adding Vectors
• When adding vectors, their directions must be taken into account
• Units must be the same
• Geometric Methods
• Use scale drawings
• Algebraic Methods
• More convenient
Adding Vectors Geometrically (Triangle Method)
• Draw the first vector A with the
appropriate length and in the
direction specified, with respect to a
coordinate system
A+ B
• Draw the next vector B with the B
appropriate length and in the
direction specified, with respect to a
whose origin is the
coordinate system
end of vector A and parallelto the
coordinate system used for A: “tip-to-
tail”. A
• The
resultant is drawn from the origin
ofA to the end of the last vector B
Adding Vectors Graphically
• When you have many vectors, just
keep repeating the process until
all are included
A+ B
• The resultant is still drawn from
the origin of the first vector to the A+ B +C
end of the last vector
A+ B
Adding Vectors Geometrically (Polygon Method)
• Draw the first vector A with the A+ B
appropriate length and in the
direction specified, with respect to
a coordinate system
• Draw the next vector B with the B
appropriate length and in the
direction specified, with respect to
the same coordinate system
• Draw a parallelogram
• The resultant is drawn as a A
diagonal from the origin
A+ B = B+ A
Vector Subtraction
• Special case of vector addition
• Add the negative of the
subtracted vector B
A − B = A + −B ( )
• Continue with standard vector
A
addition procedure −B
A− B
Describing Vectors Algebraically
Vectors: Described by the number, units and direction!
Vectors: Can be described by their magnitude and direction.
For example: Your displacement is 1.5 m at an angle of 250.
Can be described by components? For example: your
displacement is 1.36 m in the positive x direction and 0.634 m
in the positive y direction.
Components of a Vector
• A component is a part
• It is useful to use rectangular
components These are the
projections of the vector
along the x- and y-axes
Components of a Vector
• The x-component of a vector is the
projection along the x-axis
Ax
cos q = Ax = A cos q
A
• The y-component of a vector is the
projection along the y-axis
Ay
sin q = Ay = A sin q
A
• Then,
q
A = Ax + Ay A = Ax + Ay
Components of a Vector
• The previous equations are valid only if θ is measured with respect to the
x-axis
• The components can be positive or negative and will have the same units
as the original vector θ=0, A =A>0, A =0
x y
θ=45°, Ax=A cos 45°>0, Ay=A sin 45°>0
Ax < 0 Ax > 0
θ=90°, Ax=0, Ay=A>0
Ay > 0 Ay > 0
θ θ=135°, Ax=A cos 135°<0, Ay=A sin 135°>0
Ax < 0 Ax > 0 θ=180°, Ax=−A<0, Ay=0
Ay < 0 Ay < 0 θ=225°, Ax=A cos 225°<0, Ay=A sin 225°<0
θ=270°, Ax=0, Ay=−A<0
θ=315°, Ax=A cos 315°<0, Ay=A sin 315°<0
More About Components
• The components are the legs of the
right triangle whose hypotenuse is A
Ax = A cos(q )
Ay
A
Ay = A sin(q )= A x
2
+ A2
y and q = tan
−1
Ax
A = ( A )2 + (A )2
x y
Ay −1
Ay
tan (q ) = or q = tan q
Or,
Ax Ax
Unit Vectors
• Components of a vector are vectors
A = Ax + Ay
• Unit vectors i-hat, j-hat, k-hat
iˆ → x ˆj → y kˆ → z
q • Unit vectors used to specify direction
y
• Unit vectors have a magnitude of 1
• Then
j
A = Axiˆ + Ay ˆj
i
x
A Magnitude
= A x + A+ ySign
k Unit vector
z
Adding Vectors Algebraically
• Consider two vectors
A = Axiˆ + Ay ˆj
B = Bxiˆ + By ˆj
• Then
A + B = ( Ax iˆ + Ay ˆj ) + ( Bxiˆ + B y ˆj )
= ( Ax + Bx )iˆ + ( Ay + B y ) ˆj
• If C = A + B = ( Ax + Bx )iˆ + ( Ay + By ) ˆj
• so C x = AxA + B=x A xC+y =AAyy + By
Example : Operations with Vectors
❑ Vector A is described algebraically as (-3, 5), while
vector B is (4, -2). Find the value of magnitude and
direction of the sum (C) of the vectors A and B.
A = −3iˆ + 5 ˆj B = 4iˆ − 2 ˆj
C = A + B = (−3 + 4)iˆ + (5 − 2) ˆj = 1iˆ + 3 ˆj
Cx = 1 Cy = 3
2 2
C = (C x + C y )1/ 2 = (12 + 32 )1/ 2 = 3.16
Cy
q = tan −1
= tan −1 3 = 71.56
Cx
Summary
Ax = A cos(q )
Ay = A sin(q )
• Polar coordinates of vector A (A, q)
( ) ( y)
2 2
A = A + A
• Cartesian coordinates (Ax, Ay)
x
−1
Ay Ay
• Relations between them: A = Axiˆ + Ay ˆj + Az kˆ tan (q ) = or q = tan
Ax Ax
• Unit vectors:
• Addition of vectors: C = A + B = ( Ax + Bx )iˆ + ( Ay + By ) ˆj
C x = Ax + Bx C y = Ay + By
• Scalar multiplication of a vector: aA = aAxiˆ + aAy ˆj
• Multiplication of two vectors? It is possible, and we will introduce it later as it
comes up.
